Metamath Proof Explorer


Theorem sigarid

Description: Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017)

Ref Expression
Hypothesis sigar
|- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
Assertion sigarid
|- ( A e. CC -> ( A G A ) = 0 )

Proof

Step Hyp Ref Expression
1 sigar
 |-  G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
2 1 sigarval
 |-  ( ( A e. CC /\ A e. CC ) -> ( A G A ) = ( Im ` ( ( * ` A ) x. A ) ) )
3 2 anidms
 |-  ( A e. CC -> ( A G A ) = ( Im ` ( ( * ` A ) x. A ) ) )
4 cjcl
 |-  ( A e. CC -> ( * ` A ) e. CC )
5 id
 |-  ( A e. CC -> A e. CC )
6 4 5 mulcomd
 |-  ( A e. CC -> ( ( * ` A ) x. A ) = ( A x. ( * ` A ) ) )
7 cjmulrcl
 |-  ( A e. CC -> ( A x. ( * ` A ) ) e. RR )
8 6 7 eqeltrd
 |-  ( A e. CC -> ( ( * ` A ) x. A ) e. RR )
9 8 reim0d
 |-  ( A e. CC -> ( Im ` ( ( * ` A ) x. A ) ) = 0 )
10 3 9 eqtrd
 |-  ( A e. CC -> ( A G A ) = 0 )